Studying the Effect of Voltage Unbalance on the Working Characteristics of LSPMSM

2.1 Voltage unbalance

The VU is a situation where the voltage magnitude and phase angle are different [15-17]. According to NEMA, VU is defined as the line voltage unbalance rate (LVUR), which is determined as follows:

$\% L V U R=\frac{\text { max voltage deviation from the avg } L V}{\operatorname{avg} L V} .100$    (1)

where, LV-line voltage. According to IEEE standard 112 from 1991, VU is defined as the phase voltage unbalance rate (PVUR), which is determined as follows:

$\% P V U R=\frac{\text { max voltage deviation from the avg } L V}{a v g \text { phase voltage }} .100$  (2)

According to the actual definition: The actual definition of VU is defined as the ratio of the inverse and direct components of the voltage. The percentage VU factor (% VUF), or the actual definition, is determined as follows:

$\% V U F=\frac{\text { inverse voltage components }}{\text { direct voltage components }} .100$    (3)

Complex quantities are used when calculating voltage components, such as those related to VU. However, to avoid working with complex numbers, Eq. (4) is used, which approximates Eq. (3).

$\%$ voltage unbalance $=\frac{82 \cdot \sqrt{V_{a b e}^2+V_{b c e}^2+V_{c a e}^2}}{a v g L V}$   (4)

where, V_abe is the deviation value between the line voltage V_ab and the average line voltage, similarly for V_bce and V_cae.

2.2 Model of LSPMSM

The general model of the synchronous machine (SM) is presented in Figure 1 [18, 19]:

1.png

Figure 1. The general model of the SM

where, $\omega_r$ rotor speed, rotor with excitation and damper windings; w_d, w_q - armature winding of the d, q axes; u_d, u_q - winding voltages; w_cd, w_cd - damper windings of the d, q axes; w_fd, w_fq - excitation windings of d, q axes; u_f - excitation voltage.

To reduce the number of rotating electromotive force components in the equations, a general diagram is used with the armature winding placed on the rotor (Figure 2).

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Figure 2. The general model of the SM has the armature winding placed in the stator and the field winding in the rotor

The process of converting electrical to mechanical energy between the windings is independent of whether the windings are stationary or in motion; it only depends on the relative motion of those windings.

The system of equations for the SM is typically described in the d-q coordinate system. If the field is placed in the rotor, the rotor experiences a rotating synchronous magnetic field, inducing an alternating sinusoidal current in the stator winding. Based on the model in Figure 2, the set of voltage balance equations of a synchronous motor is as follows:

$\left\{\begin{array}{l}u_d=\frac{d \Psi_d}{d t}-\Psi_q \cdot \omega_r+r \cdot i_d \\ u_q=\frac{d \Psi_q}{d t}+\Psi_d \cdot \omega_r+r \cdot i_q \\ u_f=\frac{d \Psi_f}{d t}+r_f \cdot i_f \\ 0=\frac{d \Psi_{c d}}{d t}+r_{c d} \cdot i_{c d} \\ 0=\frac{d \Psi_{c q}}{d t}+r_{c q} \cdot i_{c q}\end{array}\right.$     (5)

where, i_d, i_q - armature currents of the d, q-axes; i_f – excitation current; i_cd, i_cq - damper winding currents of the d, q axes.

The flux linkages are defined by the equations:

$\left\{\begin{array}{l}\Psi_d=L_d \cdot i_d+M_d \cdot i_f+M_d \cdot i_{c d} \\ \Psi_q=L_q \cdot i_q+M_q \cdot i_{c q} \\ \Psi_f=L_f \cdot i_f+M_d \cdot i_d+M_d \cdot i_{c d} \\ \Psi_{c d}=L_{c d} \cdot i_{c d}+M_d \cdot i_d+M_d \cdot i_f \\ \Psi_{c q}=L_{c q} \cdot i_{c q}+M_d \cdot i_q\end{array}\right.$     （6）

where, L_d, L_q - armature inductances of the d, q axes; L_f - inductance of excitation winding; L_cd, L_cq - damper inductances of the d, q axes; M_d, M_q - mutual inductances of the d, q axes.

The total inductance is equal to the sum of mutual inductance and leakage inductance (L = M + L_t), assuming that the mutual fluxes of the d, q axes are linkage to all windings, while the leakage flux is linkage to only one winding. The torque, T_el is defined by the formula (based on the current):

$T_{e l}=M\left(i_f i_q+i_q i_{c d}-i_d i_{c q}\right)$    （7）

where, M = M_d = M_q with non-salient pole construction of the rotor.

Or the torque is defined by the formula (based on the magnetic flux linkage).

$T_{e l}=\Psi_d . i_q-\Psi_q . i_d$  （8）

For salient pole electric machines, an additional reactive torque component arises due to the differing magnetic permeability between the d, q axes. The set of equations representing the electromechanical conversion process of the SM is as follows:

$\left\{\begin{array}{l}u_d=L_d \frac{d}{d t} i_d+M_d \frac{d}{d t} i_f-L_d i_q \omega_r-M_q i_{c q} \omega_r+r i_d \\ u_q=L_q \frac{d}{d t} i_q+M_q \frac{d}{d t} i_{c q}+L_d i_d \omega_r+M_d i_f \omega_r+r i_q \\ u_f=L_f \frac{d}{d t} i_f+M_d \frac{d}{d t} i_d+M_d \frac{d}{d t} i_{c d}+r_f i_d \\ 0=L_{c d} \frac{d}{d t} i_{c d}+M_d \frac{d}{d t} i_d+M_d \frac{d}{d t} i_f+r_{c d} i_{c d} \\ 0=L_{c q} \frac{d}{d t} i_{c q}+M_q \frac{d}{d t} i_q+r_{c q} i_{c q}\end{array}\right.$      (9)

The damper winding only operates when the rotor is oscillating or running at a speed other than synchronous. The damper winding can be neglected, in that case:

$\left\{\begin{array}{l}u_d=L_d \frac{d}{d t} i_d+M_d \frac{d}{d t} i_f-L_d i_q \omega_r+r i_d \\ u_q=L_q \frac{d}{d t} i_q+L_d i_d \omega_r+M_d i_f \omega_r+r i_q \\ u_f=L_f \frac{d}{d t} i_f+M_d \frac{d}{d t} i_d+r_f i_d\end{array}\right.$    (10)

where, L_d = M_d + L_td, L_q = M_q + L_tq, L_f = M_d + L_tf, L_td, L_tq, L_tf - Leakage inductance of the d, q axes and inductance of the excitation winding, substituting the values from (10) into (9) yields:

$\left\{\begin{array}{l}u_d=L_{t d} \frac{d}{d t} i_d+M_d \frac{d}{d t} i_d+M_d \frac{d}{d t} i_f-L_q i_q \omega_r+r i_d \\ u_q=L_q \frac{d}{d t} i_q+M_q \frac{d}{d t} i_q+L_d i_d \omega_r+M_d i_f \omega_r+r i_q \\ u_f=L_{t f} \frac{d}{d t} i_f+M_d \frac{d}{d t}\left(i_f+i_d\right)+r_f i_f\end{array}\right.$     (11)

Combining the components with the torque balance equation, the set of equations (voltage balance, torque balance) for the SM is applicable to both steady-state and transient conditions.

$\left\{\begin{array}{l}E_q=L_q i_q \omega_r E_d=L_d i_d \omega_r+M_d i_d \omega_r \\ u_d=L_{t d} \frac{d}{d t} i_d+M_d \frac{d}{d t}\left(i_{f+} i_d\right)-E_q+r i_d \\ u_q=L_{t q} \frac{d}{d t} i_q+M_q \frac{d}{d t} i_q+E_d+r i_d \\ u_f=L_{t f} \frac{d}{d t} i_f+M_d \frac{d}{d t}\left(i_f+i_d\right)+r_f i_f \\ \sum M=J \frac{d \omega_r}{d t}=J \frac{d^2 \theta}{d t^2}\end{array}\right.$         (12)

where, J - the moment of inertia; p - the number of pole pairs.

Basically, the LSPMSM model fully inherits the equations of the general synchronous motor model [20]. The mathematical model of the LSPMSM consists of a system of equations for the stator voltage, rotor voltage, stator magnetic flux, and rotor magnetic flux, as given in formulas (13) to (16).

$\left\{\begin{array}{l}u_{d s}=r_1 i_{d s}+\frac{d \Psi_{d s}}{d t}-\omega_r \cdot \Psi_{q s} \\ u_{q s}=r_l i_{q s}+\frac{d \Psi_{q s}}{d t}+\omega_r . \Psi_{d s}\end{array}\right.$    (13)

$\left\{\begin{array}{l}u_{d r}^{\prime}=r_{d r}^{\prime} i_{d r}^{\prime}+\frac{d \Psi_{d r}^{\prime}}{d t}=0 \\ u_{q r}^{\prime}=r_{q r}^{\prime} i_{q r}^{\prime}+\frac{d \Psi_{q r}^{\prime}}{d t}=0\end{array}\right.$     (14)

$\left\{\begin{array}{l}\Psi_{d s}=\left(L_{l s}+L_{m d}\right) i_{d s}+L_{m d} i_{d r}^{\prime}+\Psi_m^{\prime} \\ \Psi_{q s}=\left(L_{l s}+L_{m q}\right) i_{q s}+L_{m q} i_{q r}^{\prime}\end{array}\right.$    (15)

$\left\{\begin{array}{l}\Psi_{d r}^{\prime}=L_{l r}^{\prime} i_{d r}^{\prime}+L_{m d}\left(i_{d s}+i_{d r}^{\prime}\right)+\Psi_m^{\prime} \\ \Psi_{q r}^{\prime}=L_{l r}^{\prime} i_{q r}^{\prime}+L_{m q}\left(i_{q s}+i_{q r}^{\prime}\right)\end{array}\right.$     (16)

where, $\omega_r$ - rotor angular velocity; $\Psi_m^{\prime}$ - stator flux generated by the PMs; L_ls - Leakage inductance of the stator winding; L_md and L_mq - magnetizing inductances of d, q axes; i_ds, i_qs - stator current of d, q axes; i'_dr, i'_qr - rotor equivalent current of d, q axes.

The equivalent circuits of LSPMSM in the mathematical model shown in Figure 3 and Figure 4.

3.png

Figure 3. The d-axis equivalent circuit of LSPMSM

4.png

Figure 4. The q-axis equivalent circuit of LSPMSM

The electromagnetic torque of the LSPMSM is determined:

$M_{e l}=\frac{3}{2} \cdot p \cdot\left[\begin{array}{l}\underbrace{\left(L_{m d} \cdot i_{d r}^{\prime} \cdot i_{q s}-L_{\text {ma }} \cdot i_{q r}^{\prime} \cdot i_{d s}\right)}_{\text {Induction Torgue }} \\ +\underbrace{\Psi_m^{\prime} \cdot i_{q s}}_{\text {ExcitationTorque }}+\underbrace{\left(L_{m d}-L_{m q}\right) \cdot i_{d s} \cdot i_{q s}}_{\text {Rehuctancetorgue }}\end{array}\right]$     (17)

Thus, the electromagnetic torque of the LSPMSM consists of three components: T_ind - the induction torque component; T_exc - the excitation torque component; T_rel - the reluctance torque component. The electromagnetic torque:

$T_{e l}=T_{i n d}+T_{e x c}+T_{r e l}$  (18)

From Eq. (17), it can be observed that the electromagnetic torque of the LSPMSM is much more complex than that of the IM. Thus, the excitation torque and reluctance torque components are equivalent to those of the synchronous reluctance motor. The LSPMSM differs in that it includes an additional induction torque component, which plays a crucial role in the starting capability of the motor [21].

Thus, the model of LSPMSM can be replaced by the IM model and the IPMSM model shown in Figure 5 [22, 23].

5.png

Figure 5. The LSPMSM model includes IM and IPMSM

2.3 VU and LSPMSM torques

In the case of VU, the torque of the IM is significantly affected [24]. At this point, the induction torque component is determined by Eq. (19) below:

$\begin{aligned} T_{I M}= & \frac{3 R_r\left(V_s^l\right)^2\left(1-K_u\right)}{\left[\left(R_r\right)^2+\left(1-K_u\right)^2\left(X_r\right)^2\right] \cdot \omega_s} \\ & -\frac{3 R_r\left(V_s^l\right)^2\left(1-K_u\right)}{\left[\left(R_r\right)^2+\left(1-K_u\right)^2\left(X_r\right)^2\right] \cdot \omega_s}\end{aligned}$  (19)

where, K_u is the coefficient affected by the VU.

Similarly, the VU also significantly affects the torque of the IPMSM [25].

$T_{P M S M}=\frac{n_p}{4} \cdot I_m \cdot \psi_f\left[\begin{array}{l}(3-k) \sin \left(\omega_e t-\frac{n_p}{2} \alpha\right) \\ +k \sin \left(\omega_e t+\frac{n_p}{2} \alpha\right)\end{array}\right]$   (20)

The LSPMSM is a combination of the IM and IPMSM, thus the analysis above indicates that the occurrence of VU significantly affects the torque characteristics as well as other working characteristics of the LSPMSM.

The experimental evaluation was conducted in the laboratory of the Faculty of Electromechanics at Hanoi University of Mining and Geology. The experimental principal diagram is shown in Figure 13.

13.png

Figure 13. Experimental principle diagram of LSPMSM

Note: 1 - Auto transformer; 2 - Voltage transformer; 3 - Current transformer; 4 - Testing load; 5 - Experimental LSPMSM;6-Rotor of LSPMSM; 7 - Encoder; 8- Displaying device

The testing process was conducted by varying the single-phase loads (load A, load B, load C), resulting in VMPAU in the voltage supplied to the LSPMSM corresponding to case 4. The testing results are measured using the KYORITSU 6310 multifunction power analyzer, which recorded several signals related to voltage, current, and power factor $\cos \varphi$. The current characteristics of phase A measured by the KYORITSU 6310 during the testing process are presented in Figure 14.

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Figure 14. Measured current characteristic of phase A of LSPMSM

The current characteristics in Figure 14 indicate that when phase VU occurs, the current of the LSPMSM is no longer sinusoidal like that of the IM. However, the LSPMSM exhibits a significant level of non-sinusoidal behavior. In addition to the non-sinusoidal level, the current characteristics of the LSPMSM show a peaked shape at the beginning of the half-cycle, indicating that the motor's magnetic circuit has saturated. The current characteristics at the end of the half-cycle exhibit a sharp peak, causing a surge in current. At this point, this can lead to demagnetization of the PM, potentially causing the LSPMSM to malfunction and resulting in motor damage.

Figure 14 also indicates that the current characteristics during the testing process closely similar to the current waveform simulated in Figure 9, additionally the measured current has a high THD_i when analyzing FFT of the current harmonic of phase A, in this case THD_i = 48.8%. This measured value is close to the THD_i value obtained from the simulation evaluation in Table 4 (THD_i = 45.41%). Similarly, the investigation of the power factor cosφ of the LSPMSM obtained during the testing process is presented in Figure 15.

Figure 15 indicates that the cosφ varies and fluctuates significantly during VU time. In the steady-state time, the $\cos \varphi$ is equal to 0.826, which is similar to the cos$\varphi$ obtained during the simulation (0.8294). Thus, it can be observed that with the limited experimental model, the investigation of the current signals and cos$\varphi$ shows that the experimental results are somewhat consistent with the theoretical and experimental values obtained.

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Figure 15. Cos$\varphi$ characteristics of LSPMSM in testing